Linear Upper Bounds on the Ribbonlength of Knots and Links

TL;DR

This paper proves that the ribbonlength of any knot or link is linearly bounded by its crossing number, providing a universal linear upper bound using grid diagrams and vertex leveling techniques.

Contribution

It establishes the first general linear upper bound on ribbonlength for all knots and links, advancing understanding of knot complexity measures.

Findings

Ribbonlength is bounded by 2.5 times the crossing number plus one.

The proof uses binary grid diagrams and bisected vertex leveling methods.

Supports the conjecture that ribbonlength grows linearly with crossing number.

Abstract

A knotted ribbon is one of physical aspect of a knot. A folded ribbon knot is a depiction of a knot obtained by folding a long and thin rectangular strip to become flat. The ribbonlength of a knot type can be defined as the minimum length required to tie the given knot type as a folded ribbon knot. The ribbonlength has been conjectured to grow linearly or sub-linearly with respect to a minimal crossing number. Several knot types provide evidence that this conjecture is true, but there is no proof for general cases. In this paper, we show that for any knot or link, the ribbonlength is bounded by a linear function of the crossing number. In more detail, $$ \text{Rib}(K) \leq 2.5 c(K)+1. $$ for a knot or link $K$. Our approach involves binary grid diagrams and bisected vertex leveling techniques.